Modal Model Theory

Modal model theory injects ideas from modal logic into the traditional subject of model theory in mathematical logic. On this view, one treats the class of all models Mod(T) of some first-order theory T as a Kripke model of possible worlds. Worlds, here, are structures in Mod(T), and a statement s is possible at model M just in case it extends to another model whereat s is true. Similarly, statement s is necessary at model M if all its extensions satisfy s. This research includes an effort presented in [1], where we introduce the subject of modal model theory and relevant fundamental theorems. We demonstrate the importance of this subject by means of an example -- modal graph theory -- a particularly insightful case illustrating the remarkable power of the modal vocabulary. The modal language of graph theory can express connectedness, k-colourability, finiteness, countability, size continuum, size aleph-one, aleph-two, aleph-omega, beth-omega, first beth-fixed point, first beth-hyper-fixed-point, and much more. A graph obeys the maximality principle -- every possibly necessary statement is already true -- with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs.

For clarity I should like to separate distinct but closely related languages arising in modal model theory.

1) We denote by L the language of the theory T.
2) L' is the closure of L under the modal operators of possibility and necessity as well as Boolean connectives (but not quantifiers).
3) L'' is the full first-order modal language, closing L under modal operators, Boolean connectives and quantifiers.
4) L''@ extends the full modal language with the actuality operator @, which allows one to refer to the actual world.

While it is hard to predict the direction in which this project might develop, my intention is to further analyse modal theories arising in the context of these languages. For example, in [1], we note that many properties of the intermediate modal language L' do not extend to the full modal languages L'' or L''@ -- this led us to multiple questions that I should like to look at. In particular:

Question 1. In ZFC, can one define the satisfaction relation for modal graph theory for the class of all graphs?
Question 2. Is actuality @ expressible in modal graph theory?
Question 3. Is transcendence degree expressible in modal field theory?
Question 4. Can one generalise Ehrenfeucht-Fraïssé games to the full modal language, with actuality, L''@?

My research, however, shall not be limited to issues arising in [1] and sole problem-solving. I view this project not only as a contribution to the field of mathematical logic but also, for example, as an attempt to join the discussion surrounding the debate on pluralism in the philosophy of mathematical foundations. Indeed, modal perspective on model theory has its roots in works such as [2] and [3], which exemplify the philosophical perspective that there is no one true and ultimate mathematical universe; rather there are many -- all comprising the mathematical multiverse as a whole.

This project falls within the EPSRC Logic and combinatorics research area.